Thesis

Degree for the central curve of quadratic programming

Our research begins with a standard quadratic programming problem from the field of optimization. One family of solution methods is called interior point methods. These methods trace a path through the feasible region until they converge to the optimal solution. This path, defined by polynomial equations, is a piece of the central curve. Recent work has been done studying the degree and total curvature of the central curve in the linear programming case. We broaden the scope and look to prove similar results in the quadratic case. Through computer experimentation we hypothesized that the degree can be considered just for the case that the objective function has a generic diagonal matrix. We prove the reduction to diagonal and consider the degree of the central curve only in this case. We proceed by constructing a monomial ideal resulting from the optimality conditions of our quadratic program. We prove the degree of this monomial ideal, and show that this degree is an upper bound for the degree of the quadratic central curve. While the proposed degree is in fact exact, we only prove an upper bound for this thesis. Our result has nice symmetry with the main result of its linear counterpart.

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